Spectral dynamics for the infinite dihedral group and the lamplighter group

TL;DR

This paper explores the spectral properties of the infinite dihedral and lamplighter groups through projective spectra, rational maps, and Julia sets, revealing new relationships and invariants in group $C^*$-algebras.

Contribution

It establishes a precise connection between the projective spectrum and Julia sets for these groups, improving understanding of their spectral dynamics.

Findings

The projective spectrum equals the Julia set for the infinite dihedral group.

The extended pencil approach simplifies the relationship between spectra and Julia sets.

For the lamplighter group, the Julia set coincides with the extended indeterminacy set.

Abstract

For a tuple $A=(A_0,A_1,\cdots,A_n)$ of elements in a Banach algebra $\mathfrak{B}$, its projective (joint) spectrum $p(A)$ is the collection of $z\in \mathbb{P}^n$ such that $A(z)=z_0A_0+z_1A_1+\cdots+z_nA_n$ is not invertible. If $\mathfrak{B}$ is the group $C^*$-algebra for a discrete group $G$ generated by $A_0, A_1,\dots, A_n$ with a representation $\rho$, then $p(A)$ is an invariant of (weak) equivalence for $\rho$. In \cite{BY}, B. Goldberg and R. Yang proved that the Julia set $\mathcal{J}(F)$ of the induced rational map $F$ for the infinite dihedral group $D_\infty$ is the union of the projective spectrum with the extended indeterminacy set. But the extended indeterminacy set $E_F$ is complicated. To obtain a better relationship between the projective spectrum and the Julia set, by replacing $A_\pi(z)=z_0+z_1\pi(a)+z_2\pi(t)$ with the extended pencil $A_\pi(z)=z_0+z_1\pi(a)+z_2\pi(t)+z_3\pi(at)$, where $\pi$ is the Koopman representation, and using the method of operator recursions, we show that $p(A_\pi)=\mathcal{J}(F).$ Further, we study the spectral dynamics for the Lamplighter group $\mathcal{L}$, and prove that $\mathcal{J}(Q)=E_Q$, where $Q$ is the rational map associated with $\mathcal{L}$.